Andrew J. Dean, Department of Mathematical Sciences, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario, Canada P7B 5E1; e-mail: ajdean@lakeheadu.ca

Sarah K. Lucky , Department of Mathematical Sciences, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario, Canada P7B 5E1; e-mail: sklucky@lakeheadu.ca

Abstract/Résumé:

It is shown that if \(G\) is a lattice ordered countable group, then every action of \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) on \(G\) arises as an inductive limit of actions of \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) on simplicial groups. Some parts of the argument work in greater generality, and are proved for general finite abelian groups. A template is given for proving similar results for other such groups.

On montre que si \(G\) est un groupe dénombrable treillis-ordonné, alors toute action de \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) sur \(G\) provient d’une limite inductive d’actions de \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) sur des groupes simpliciaux. Des parties de cet argument fonctionnent dans une généralité plus grande et sont prouvées pour des groupes abéliens finis en général. Un modèle est donné pour prouver des résultats similaires pour d’autres groupes de ce type.

Keywords: Dimension groups, K-theory, classification

AMS Subject Classification: Classifications of $C^*$-algebras; factors, Noncommutative dynamical systems 46L35, 46L55

PDF(click to download): Actions of $({Z}/2{Z})$ x $({Z}/2{Z})$ on Lattice Ordered Dimension Groups

Bit Na Choi, Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA; e-mail: Bitna.Choi@unh.edu

Andrew J. Dean, Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON, Canada P7B 5E1; e-mail: ajdean@lakeheadu.ca

Abstract/Résumé:

We show that a \(\mathbb{Z}_2\) action on a lattice-ordered dimension group will arise as an inductive limit of \(\mathbb{Z}_2\) actions on simplicial groups. The motivation for this study is the range of invariant problem in Elliott and Su’s classification of AF type \(\mathbb{Z}_2\) actions. We modify the proof of the Effros-Handelman-Shen theorem to include \(\mathbb{Z}_2\) actions.

Nous montrons qu’une action de \(\mathbb{Z}_2\) sur un groupe de dimension ordonné par treillis apparaît comme une limite inductive d’actions de \(\mathbb{Z}_2\) sur des groupes simpliciaux. La motivation de cette étude est le problème de la gamme de l’invariant dans la classification d’Elliott et de Su des actions de \(\mathbb{Z}_2\) de type AF. Nous modifions la preuve du théorème d’Effros-Handelman-Shen pour inclure les actions de \(\mathbb{Z}_2\).

Keywords: Dimension groups, K-theory, classification

AMS Subject Classification: Ordered abelian groups; Riesz groups; ordered linear spaces, $K_0$ as an ordered group; traces 06F20, 19K14

PDF(click to download): A Modification of the Effros-Handelman-Shen Theorem with ${Z}_2$ actions

George A. Elliott,Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

It is shown that the index map in the theory of real Banach algebras is unique as a natural transformation, up to an integral multiple, and modulo a (unique) two-torsion “ghost” map arising from the order-two K\(_1\)-group of the Banach algebra \({\mathbb R}\) (of real numbers). (In the earlier paper this was shown for complex Banach algebras, of course without the “ghost” map, but in way—using Bott periodicity to pass to the opposite parity—that is not available for real Banach algebras. The present approach yields a new proof in the complex case.)

On démontre que l’application index dans la K-théorie des algèbres de Banach réelles (ou complexes) est essentiellment unique.

Keywords: K-theory, index theory

AMS Subject Classification: Index theory, K-theory and operator algebras -including cyclic theory 19K56, 46L80

PDF(click to download): Uniqueness of the Index Map in Banach Algebra K-theory, II

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

It is shown that the index map in Banach algebra K-theory, as a natural map from the K\(_1\)-group of a quotient of a Banach algebra to the K\(_0\)-group of the corresponding ideal, is unique (up to an integral multiple).

Il est démontré que l’application index dans la K-théorie des algèbres de Banach est unique, dans un sens très naturel.

Keywords: K-theory, index theory

AMS Subject Classification: Index theory 19K56

PDF(click to download): Uniqueness of the Index Map in Banach Algebra K-Theory

Sandro Molina-Cabrera, Department of Mathematics, Rıo Piedras Campus, University of Puerto Rico, Box 23355, San Juan, Puerto Rico 00931-3355, USA; email: smolinacabrera@gmail.com

Abstract/Résumé:

We show that the \(K_0\)-group of an inductive limit of recursive subhomogeneous algebras with compact metrizable spaces of dimension at most one as local spectra is torsion free. This result implies that the \(K_0\)-group of a unital simple AH algebra which is the inductive limit of recursive subhomogeneous algebras, with compact metrizable spaces of dimension at most one as local spectra, is torsion free. This proves that Li’s reduction theorem for the dimension of the local spectra of unital simple AH algebras cannot be improved, in other words, that the dimension of the local spectra of unital simple AH algebras cannot be further reduced from two to one, even when we use subhomogeneous algebras. This also shows that if a reduction theorem for the dimension of the local spectra of simple inductive limits of recursive subhomogeneous algebras exists, then, after the reduction, the local spectra of the building blocks cannot always be one dimensional.

Nous démontrons que le \(K_0\)-groupe d’une limite inductive des algèbres sous-homogènes récursives, dont les spectres locaux consistent en des espaces compacts métrisables de dimension au plus un, n’a pas de torsion. Ce résultat implique que les \(K_0\)-groupes d’une algèbre AH simple et avec l’unité qui est la limite des algèbres sous-homogènes rećursives, dont les spectres locaux consistent en des espaces compacts métrisables de dimension au plus un, n’a pas de torsion. Cela prouve que le théorème de Li de la réduction pour la dimension des spectres locaux des algèbres AH simples et avec l’unité ne peut pas être améliorée, en d’autres termes, que la dimension des spectres locaux des algèbres AH simples et avec l’unité ne peut pas encore être réduit de deux à un, même quand on utilise des algèbres sous-homogènes. Cela montre aussi que si un théorème de réduction pour la dimension des spectres locaux d’une limite inductive simple des algèbres sous-homogènes récursives existe, alors, après la réduction, les spectres locaux des blocs de construction ne peuvent pas être toujours de dimension un.

Keywords: AH algebras, C*-algebra, K-theory, classification, recursive subhomogeneous algebras

AMS Subject Classification: K-theory and operator algebras -including cyclic theory 46L80

PDF(click to download): Torsion in the ${K_0}$-Group of a Recursive Subhomogeneous Algebra

S.G. Walters

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords: C*-algebra, Chern characters, K-theory, automorphisms, rotation algebras, unbounded traces

AMS Subject Classification: $K_0$ as an ordered group; traces, Automorphisms, K-theory and operator algebras -including cyclic theory 19K14, 46L40, 46L80

PDF(click to download): On the irrational quartic algebra